Abstract

With Quantum Singular Value Transformation (QSVT) emerging as a unifying framework for diverse quantum speedups, efficient construction of block encodings—its fundamental input model—has become increasingly crucial. However, devising explicit block-encoding circuits has remained a significant challenge. A general strategy is the Linear Combination of Unitaries (LCU) method, though its practical utility is often limited by substantial gate overhead. To address this, we introduce the Fast One-Qubit-Controlled Select LCU (FOQCS-LCU), a compact formulation that requires only a linear number of ancilla qubits and admits explicit decompositions into one- and two-qubit gates. By exploiting Hamiltonian structure, we design a parametrized family of efficient Dicke-state preparation routines that enable systematic construction of state-preparation oracles at greatly reduced gate cost. The check-matrix formalism further yields a constant-depth SELECT oracle implemented as two fully parallelizable layers of singly controlled Pauli gates. We also present explicit block-encoding circuits for matrix polynomial transformations and show that the additional circuit-depth overhead scales linearly in the polynomial degree d, independent of system size or the cost of encoding the original matrix alone. Moreover, both the FOQCS -LCU circuits and their associated polynomial transformations can be controlled with negligible overhead, enabling efficient applications such as Hadamard tests. We construct explicit implementations for representative spin models, including Heisenberg and spin-glass Hamiltonians, and provide detailed non-asymptotic gate counts. Numerical benchmarks demonstrate over an order-of-magnitude reduction in CNOT count compared to conventional LCU approaches, establishing a practical route to low-depth block encodings for a broad class of structured matrices.